r²SCAN-Based Double-Hybrids

Functional Implementation

• Pr²SCAN functionals can be build in Orca and Turbomole via the LibXC
• r²SCAN can be used with libxc version 5.1.0 as MGGA_X_R2SCAN (id=497), MGGA_C_R2SCAN (id=498)
• XCFun 2.1.0 also implements support for the r²SCAN functional

• Routines for r²SCAN functional implementation can be found at https://gitlab.com/dhamil/r2scan-subroutines (Fortran), routines for Vasp are included

• $\kappa$ Pr²SCAN functionals employ the regularized MP2 implemented in a development version of ORCA 5.0

• $\omega$ Pr²SCAN functionals utilize the error function splitting of the two-electron operator implemented in a development version of ORCA 5.0

• Libraries implementing DFT-D4 can be found at https://github.com/dftd4/dftd4 (Fortran) and https://github.com/dftd4/cpp-d4 (C++)

• NL dispersion correction DFT-NL (VV10, by Vydrov and Van Voorhis) can be used in Orca and Turbomole. For program-suits that do not support the usage of a scaling factor of the NL energy, one can scale the obtained NL energy by $NLScal$ manually.

ORCA Input Files

Double-hybrids: - r²SCAN0-DH-D4 - r²SCAN-CIDH-D4 - r²SCAN-QIDH-D4 - r²SCAN0-2-D4

• Pr²SCAN50-D4 / NL

• Pr²SCAN69-D4 / NL

• kPr²SCAN50-D4 / NL

• wPr²SCAN50-D4 / NL

Range-separated hybrid: - wr²SCAN-D4 / NL

Functional Parameters

Functional

All functionals employ the following fundamental relation.

$E{XC}^{DH} = (1-a{X})E{X}^{DFT} + a{X}E{X}^{HF} + (1-a{C})E{C}^{DFT} + a{C}(a{SS}E{C}^{SS-MP2}+a{OS}E{C}^{OS-MP2})$

| Functional | $aX$ | $aC$ | $a{OS}$ | $a{SS}$ | $\kappa$ | $\omega$ | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | r²SCAN0-DH | $1/2$ (0.50) | $1/8$ (0.13) | $4/3$ | 0 | | | | r²SCAN-CIDH | $6^{-1/3}$ (0.55) | $1/6$ (0.17) | $4/3$ | 0 | | | | r²SCAN-QIDH | $3^{-1/3}$ (0.69) | $1/3$ (0.33) | $4/3$ | 0 | | | | r²SCAN0-2 | $2^{-1/3}$ (0.79) | $1/2$ (0.50) | $4/3$ | 0 | | | | Pr²SCAN50 | $1/2$ (0.50) | $1/4$ (0.25) | $4/3$ | 0 | | | | Pr²SCAN69 | $3^{-1/3}$ (0.69) | $4/9$ (0.44) | $4/3$ | 0 | | | | $\kappa$ Pr²SCAN50 | $1/2$ (0.50) | $3/10$ (0.30) | $4/3$ | 0 | 2.7500 | | | $\omega$ Pr²SCAN50 | $1/2$ (0.50) | $7/20$ (0.35) | $4/3$ | 0 | | 0.2140 |

Dispersion Correction

Dispersion correction parameters were fitted against the S22x5, NCIBLIND10, and S66x8 (2022 revision by Martin et al.) benchmark sets for non-covalent interaction energies by least-squares Levenberg-Marquardt minimization. All parameters were fitted close to the complete basis set limit using the large def2-QZVPP quadruple-zeta basis set.

D4

| Functional | $s6$ | $s8$ | $s9$ | $a1$ | $a_2$ | | :---: | :---: | :---: | :---: | :---: | :---: | | r²SCAN0-DH | 0.9424 | 0.3856 | 1.0000 | 0.4271 | 5.8565 | | r²SCAN-CIDH | 0.8666 | 0.5336 | 1.0000 | 0.4171 | 5.9125 | | r²SCAN-QIDH | 0.7867 | 0.2955 | 1.0000 | 0.4001 | 5.8300 | | r²SCAN0-2 | 0.7386 | 0.0000 | 1.0000 | 0.4030 | 5.5142 | | Pr²SCAN50 | 0.7964 | 0.3421 | 1.0000 | 0.4663 | 5.7916 | | Pr²SCAN69 | 0.7167 | 0.0000 | 1.0000 | 0.4644 | 5.2563 | | $\kappa$ Pr²SCAN50 | 0.8402 | 0.1212 | 1.0000 | 0.4382 | 5.8232 | | $\omega$ Pr²SCAN50 | 0.8143 | 0.3842 | 1.0000 | 0.4135 | 5.8773 |

This Markdown table includes the provided data with appropriate formatting for GitHub. Adjustments

NL

Non-local dispersion correction makes use of a scaling factor $a{NL}$, that is given by $a{NL}=1-a_C$.

| Method | $a{NL}$ | $NL{b}$ | | :---: | :---: | :---: | | Pr²SCAN50 | 0.7500 | 10.9207 | | Pr²SCAN69 | 0.5556 | 9.0691 | | $\kappa$ Pr²SCAN50 | 0.7000 | 10.6723 | | $\omega$ Pr²SCAN50 | 0.6500 | 9.4149 |